Minimum Cost to Buy Apples

Minimum Cost to Buy Apples

What is this problem about?

In the Minimum Cost to Buy Apples problem, you are given a set of cities connected by roads with specific travel costs. Each city also has a price for an apple. You want to find, for each city i, the minimum cost to start at i, travel to any city j (including i), buy an apple at city j, and return to city i. The travel cost is usually multiplied by a constant (like 2 for a round trip).

Why is this asked in interviews?

This Minimum Cost to Buy Apples interview question is seen at companies like Media.net and Directi. It is a multi-source shortest path problem. It tests if a candidate can adapt Dijkstra's Algorithm to handle multiple possible "targets" (apple sources) simultaneously or use a "super-node" trick.

Algorithmic pattern used

The Multi-Source Dijkstra interview pattern is the most efficient.

1. Instead of running Dijkstra from every city, imagine a "virtual source" node.
2. Connect this virtual source to every city i with an edge weight equal to the apple_cost[i].
3. Run a single Dijkstra starting from this virtual source.
4. The distance to any city i in this graph will be the minimum cost to buy an apple and "get to" city i.
5. Note: Since you need a round trip from i to j, the road weights in your graph should be 2 * travel_cost.

Example explanation

City 1: apple=10. City 2: apple=5. Road 1-2: cost 2.

1. Round trip cost 1-2-1 = 2 * 2 = 4.
2. To buy at 1: cost = 10.
3. To buy at 2 starting from 1: travel(1->2) + apple(2) + travel(2->1) = 2 + 5 + 2 = 9.
4. Min cost for City 1 is 9.
5. Min cost for City 2 is 5 (buy locally).

Common mistakes candidates make

• Running Dijkstra N times: Running a full Dijkstra from every single city ($O(N \cdot E \log V)$). This will likely time out for large graphs.
• Forgetting the return trip: Only calculating the cost to get to the apple, forgetting that you must return to the starting city.
• Ignoring local purchase: Forgetting that the cheapest apple might be in the city where you already are.

Interview preparation tip

The "Super-Source" or "Virtual Node" is a common trick in Graph interview patterns. If you have multiple starting points or multiple potential destinations and want the "closest" one, try connecting them all to a single virtual node with zero or cost-based weights.